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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

16857 questions
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Why does $k[X,Y]/(XY)$ have two minimal primes?

I working on a problem for practice. For $k$ a field, I was able to show that any element of $A=k[X,Y]/(XY)$ has a unique representation in form $a+f(X)X+g(Y)Y$ for $a\in k$, $f(X)\in k[X]$ and $g(Y)\in k[Y]$. Why does $k[X,Y]/(XY)$ have exactly two…
Hailie Mathieson
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Remark on the dimension of quotient by prime homogeneous ideal.

There's a remark near the end of a section I'm reading about Hilbert polynomials that I don't fully understand. Let $K$ be a field, and let $A=K[X_0,\dots,X_N]$ be a polynomial ring, which is graded in the standard way (the elements of degree $n$…
Buble
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Proving one form of Hilbert's Nullstellensatz

I have been trying to prove the following problem in Atiyah Macdonald concerning one form of Hilbert's Nullstellensatz. The problem is as follows: If $X$ is an affine algebraic variety (the set of all points satisfying $f_a(t_1, \ldots, t_n) = 0$,…
user38268
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characterizing free modules by exterior power

Assume $M$ is a (finitely generated) $A$-module such that $\wedge^n M$ is free of rank $1$ for some $n \geq 1$. Does it follow that $M$ is free of rank $n$? Or at least locally free of rank $n$? In general, is there a way of characterizing local…
Martin Brandenburg
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Derivation (Matsumura: Commutative algebra)

I am reading Masumura, Commutative algebra, Chapter 10: Derivation. The following is in pages 177, 178. Two extensions $(C, \varepsilon, i)$ and $(C_1, \varepsilon_1, i_1)$ are said to be isomorphic if there exists a ring homomorphism $f: C \to C_1$…
Pham Hung Quy
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If $P \in \operatorname{Supp}(M)$ prove that $P$ contains a prime ideal $Q$ with $Q \in \operatorname{Ass}_R(M)$.

My problem is below, Let $M$ be an $R$-module. The set of prime ideals $P$ of $R$ for which the localization $M_P$ is nonzero is called the support of $M$, denoted $\operatorname{Supp}(M)$. The set of prime ideals $Q$ of $R$ for which $Q$ is an…
ljh8372
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a subtle detail in the proof of Theorem 3.3.7 of Bruns and Herzog

Let $\phi: (R,m,k) \rightarrow (S,n,l)$ be a local homomorphism of Artinian rings, with $k,l$ being the corresponding residue fields. Let $E_R(k)$ be the injective hull of $k$ over $R$ and $E_S(l)$ the injective hull of $l$ over $S$. Suppose that we…
Manos
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Show that $(A[X])^\times=A^\times+nil(A[X])$

I am currently reading through the book Basic Commutative Algebra, by Balwant Singh, wherein the exercise I.XVI reads like: Show that $(A[X])^\times=A^\times+nil(A[X])$. Here, for a ring $A$, $A^\times$ means the set of all units, and…
awllower
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still torsion-free after nilredution?

This seems like it ought to be true and easy, but somehow I'm stymied. Let $A$ be a commutative ring (Noetherian if you like) and let $M$ be a finitely generated $A$-module. Suppose that $M$ is torsion-free in the sense that $M$ is without torsion…
Confused
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Proof of the Auslander-Buchsbaum formula in Matsumura

There is a proof of Auslander-Buchsbaum formula in Matsumura's Commutative Ring Theory page 155. I am trying to understand the case $\operatorname{pd} M = 1$. He says take a short exact sequence $$ 0 \to A^{\oplus m} \stackrel{\varphi}{\to}…
Lynn16
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Non-zero prime ideals are maximal in the ring of algebraic integers

Let $A= \{y \in \mathbb{C} :$ $y$ integral over $\mathbb{Z}$ }. Let $P\not=\{0 \}$ be a prime ideal of $A$. I am supposed to prove that $P$ is also a maximal ideal. But I cant make it, is this really even true?
user117449
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An example of Noether normalization

Let $A=k[x_1,x_2]/(x_2^2-x_1^3+x_1)$. As an example of Noether normalization, determine elements $y_1,\ldots,y_m\in A$, algebraically independent over $k$, such that $A$ is a finite $k[y_1,\ldots,y_m]$-algebra. This is a problem in the Klaus…
Jacob Ralph
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If $0\to M'\to M\to M''\to 0$ is exact, why does $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup \operatorname{Ass}(M'')$.

I'm stuck on a proof I'm reading. Let $0\to M'\stackrel{\mu}\to M\stackrel{\sigma}\to M''\to 0$ be a sequence of $A$-modules. Then $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup \operatorname{Ass}(M'')$. Let…
BTY
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$\operatorname{MaxSpec}(A)$ closed

If $A$ is an arbitrary commutative ring, is $\operatorname{MaxSpec}(A)$ closed as a subset of $\operatorname{Spec}(A)$? I wanted to think of a counterexample, but so far without success. I tried to consider generic points, but if…
Steffi
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Artinian local $k$-algebra

I encountered the term "Artinian local $k$-algebra," where $k$ is a field. I think the author meant an Artinian local ring which is a $k$-algebra, but is it by any chance equivalent to a local ring which is an Artinian $k$-algebra? Since Artian,…
ashpool
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